How Do Commutators Influence Eigenvalues in Quantum Mechanics?

[Minakami]

Homework Statement

Homework Equations

The Attempt at a Solution

I think I have to use the fact that [a+ , a] = 1 but I don't know where to apply this.

 

[gabbagabbahey]

I think I have to use the fact that [a+ , a] = 1 but I don't know where to apply this.

If $\left[\hat{a}^{\dagger},\hat{a}\right]=1$, what is $$\left[\hat{b}^{\dagger},\hat{b}\right]$$? What are $$\left[H,\hat{b}\right]$$ and $$\left[\hat{H},\hat{b}^{\dagger}\right]$$?

What are $$\hat{H}\left(\hat{b}|\psi_E\rangle\right)$$ and $$\hat{H}\left(\hat{b}^{\dagger}|\psi_E\rangle\right)$$...What does that tell you?

If $$E_0\hat{b}^{\dagger}\hat{b}|\psi_E\rangle-\frac{E_1^2}{E_0}|\psi_E\rangle=E|\psi_E\rangle$$, what is $$\langle\psi_E|\hat{b}^{\dagger}\hat{b}|\psi_E\rangle$$? Note that in any Hilbert space, an inner product is always greater than or equal to zero...what does that tell you about $E$?